A superconducting magnet is an electromagnet formed from coils of a superconducting material. As the magnet coils have zero resistance, superconducting magnets can carry high currents with zero loss (though there will be some losses from non-superconducting components), and can therefore sustain high fields with much lower energy losses than conventional electromagnets.
Superconductivity only occurs in certain materials, and only at low temperatures. A superconducting material will behave as a superconductor in a region defined by the critical temperature of the superconductor (the highest temperature at which the material is a superconductor in zero magnetic field) and the critical field of the superconductor (the highest magnetic field in which the material is a superconductor at 0K). The temperature of the superconductor and the magnetic field present limit the current which can be carried by the superconductor without the superconductor becoming resistive.
Broadly speaking, there are two types of superconducting material. Low temperature superconductors (LTS) have critical temperatures below 30K-40K, and high temperature superconductors (HTS) have critical temperatures above 30K-40K.
As the magnets require cooling to low temperatures, they are typically contained within a cryostat designed to minimise heating of the magnet. Such a cryostat typically comprises a vacuum chamber to minimise heating by convection or conduction, and may comprise one or more heat shields at temperatures intermediate between the temperature of the magnet and the external temperature to minimise heating by radiation.
The magnet itself is further cooled either by immersion in a liquid with a low boiling point (such as liquid nitrogen (77K), or liquid helium (4K)), or by circulating a coolant through the magnet and a cryocooler.
Efficient cooling is particularly important in applications where there are external heat sources. For example, in a nuclear fusion reactor such as a spherical tokamak, the fusion reactor generates a huge amount of heat, which means that the cooling system for the magnets must mitigate the heat production to keep the magnets at the correct operating temperature. Certain magnet geometries or constructions may also have regions which generate more heat than the rest of the magnet (e.g. joints in the superconductor).